Evaluate the following integral:

re-writing the given equation as

$\int \frac{1+\frac{9}{x^{2}}}{x^{2}+\frac{81}{x^{2}}} d x$

$\int \frac{1+\frac{9}{x^{2}}}{\left(x-\frac{9}{x}\right)^{2}+18} d x$

Let $x-\frac{9}{x}=t$

$\left(1+\frac{9}{x^{2}}\right) d x=d t$

$\int \frac{d t}{t^{2}+18}$

Using identity $\int \frac{1}{x^{2}+1} d x=\arctan (x)$

$\frac{1}{3 \sqrt{2}} \arctan \left(\frac{t}{3 \sqrt{2}}\right)+c$

Substituting $\mathrm{t}$ as $\mathrm{x}-\frac{1}{\mathrm{x}}$

$\frac{1}{3 \sqrt{2}} \arctan \left(\frac{x-\frac{1}{x}}{3 \sqrt{2}}\right)+c$